We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

中文翻译：

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分数 HJB 方程数值近似的精确误差范围


我们证明了分数和非局部 Hamilton-Jacobi-Bellman 方程的单调近似方案的精确收敛率。我们考虑文献中的扩散校正差分求积方案和基于离散拉普拉斯幂的新近似，近似是（形式上）分数阶和二阶方法。在数值分析中众所周知，收敛速度取决于解的规律性，在这里我们考虑具有不同解规律性的情况：（i）具有 Lipschitz 解的强退化问题和（ii）我们证明解有界的弱非退化问题(1,2)$ 阶 $\sigma \ 的分数阶导数。我们的主要结果是最优误差估计，其收敛速度精确地捕获了方案的分数阶和解的分数正则性。对于强简并方程，这些速率可以改善早期结果。对于大于一阶的弱非退化问题，结果是新的。在这里，我们展示了与强退化情况相比改进的速率，速率始终优于 $\mathcal{O}\big (h^{\frac{1}{2}}\big )$。